Metasurface for Engineering Superimposed Ince‐Gaussian Beams

Ince‐Gaussian beams (IGBs) are the third complete family of exact and orthogonal solutions of the paraxial wave equation and have been applied in many fields ranging from particle trapping to quantum optics. IGBs play a very important role in optics as they represent the exact and continuous transition modes connecting Laguerre–Gaussian and Hermite–Gaussian beams. The method currently in use suffers from the high cost, complexity, and large volume of the optical system. The superposition of IGBs can generate complicated structured beams with multiple phase and polarization singularities. A metasurface approach is proposed to realizing various superpositions of IGBs without relying on a complicated optical setup. By superimposing IGBs with even and odd modes, multiple phase, and polarization singularities are observed in the resultant beams. The phase and polarization singularities are modulated by setting the initial phase in the design and controlling the incident linear polarization. The compactness of the developed metasurface devices and the unique properties of the generated beams have the potential to impact many practical applications such as particle manipulation, orbital angular momentum spectrum manipulation, and optical communications.


Introduction
Laguerre-Gaussian beams (LGBs) and Hermite-Gaussian beams (HGBs) are two complete families of exact solutions for paraxial wave equation (PWE) in cylindrical and cartesian coordinates, respectively. [1]nce-Gaussian beams (IGBs), the third complete family of exact and orthogonal solutions of PWE in elliptical coordinates, are considered as generalized beams and represent the exact and continuous transition modes connecting LGBs and HGBs. [2]GBs represent a class of light beams characterized by both radial and elliptical symmetries.The typical features of these beams include elliptical symmetry, dependence on Ince parameters, orthogonality and completeness, and specific propagation characteristics. [3]For example, the ellipticity parameter in their mathematical representation can be used to shape the beam profile, which can be used to tailor the beams for various optical systems.IGBs have been used in many fields, including optical mode conversion, [4] particle trapping, [5] cell biology, [6] and optical storage. [7]IGBs were first experimentally demonstrated in a stable solid-state laser resonator, [1,8] but this approach suffers from the fixed mode order since the resonator could only be tuned for a specific order.An accurate adjustment of the laser cavity is necessary for realizing different modes. [9]The superposition of IGBs can produce complicated structured beams with multiple phase and polarization singularities, which have been applied in many fields ranging from cold atoms trapping [10] to quantum optics. [11]However, the current system for the IGB generation includes many bulky optical elements (e.g., Dove prism, SLMs, DMDs, pinhole filters, polarizers, lenses, and dichroic mirrors) and has a low resolution, [9,[12][13][14][15] increasing expense and making them impractical for applications where lightweight and compactness are needed.[18] Hence, there is an urgent need for a simple, compact, and efficient approach to generating and manipulating IGBs.
Optical metasurfaces have provided a desirable approach to manipulating light's polarization, phase, and amplitude at subwavelength scale, [19] which has been extensively used to develop a plethora of ultrathin optical devices with unusual functionalities, including spectrometers, [20,21] polarization detectors [22] and cameras, [23,24] edge imagers, [25][26][27] and longitudinal waveplates. [28,29]Recently, metasurfaces have been utilized to generate various structured beams, e.g., polarization knots, [30,31] composite vortex beams, [32,33] grafted vector vortices, [34,35] orbital angular momentum (OAM) holograms, [36,37] Airy beams, [38,39] and spatiotemporal light fields. [40,41]To tackle the fundamental and technical challenges in the generation of IGBs, we propose and experimentally demonstrate a compact metasurface platform for generating and manipulating IGBs.For the first time, a single metasurface is used to generate superimposed IGBs through the superposition of even and odd Ince modes without relying on a complicated optical setup.The resultant beams with the same polarization state feature multiple phase singularities, which are modulated by incorporating an initial phase shift in the design.The number of phase singularities is manipulated with different combinations of even and odd modes.Multiple polarization singularities are observed in the resultant vector beams, which are dynamically modulated by controlling the polarization direction of the incident light, performing an optical task that is extremely challenging or impossible with conventional optics.The compactness of the developed metasurface devices and unique properties of superimposed IGBs with multiple phase and polarization singularities render this technology very attractive for applications in many fields such as particle manipulation, OAM spectrum manipulation, and optical communications.

Results
Figure 1 illustrates the schematic of the proposed metasurface for realizing the superposition of IGBs.When a light beam with right circular polarization (RCP) shines on the metasurface, the superposition of IGBs with even and odd modes and the same left circular polarization (LCP) state is realized.The resultant beam possesses multiple phase singularities, and the total number of singularities is manipulated with different combinations of even and odd modes.Under the illumination of a linearly polarized (LP) light beam, the resultant beam with multiple phase and polarization singularities is produced through the orthogonal superposition of circularly polarized IGBs.Such a beam is the extension of the scalar IGBs and integrates polarization attributes into its spatial profile.The vector nature is revealed through the modulated intensity profile after the beam passes through a linear polarizer (analyzer).By continuously controlling the incident linear polarization, the polarization profile of the generated vector beam is dynamically modulated.
IGBs are the exact and orthogonal solutions of a PWE in an elliptical coordinate system.Along the light propagation direction (z), the elliptical coordinates can be expressed as [1] x = f (z) cos h ( ) cos () (1) Here  ∈ (0, ∞) and  ∈ (0, 2) are radial and angular elliptic variables, respectively.The curves of the confocal ellipse  and confocal hyperbola  are given as [1] x where is the semifocal separation.Here, f o and  o are the semifocal length and the Gaussian beam width at the plane z = 0, respectively.
) 1∕2 is the Gaussian beam width at an arbitrary z plane, where z r = k 2 o is the Rayleigh range and k is the wave number.In the elliptical coordinate system, IGBs with order p, degree m, and elliptic parameter o can be written as [2] IGB e p, m (r, ) = where C and S are the constants and r is the radial distance in a z-plane.The superscripts "e" and "o" denote the even and odd modes of IGBs, respectively.C m p (n, ) and S m p (n, ) are the even and odd Ince polynomials with order p, degree m and ellipticity , .On the other hand, the indices of HGB e,o n x , n y for even IGBs are given as n x = m and n y = p − m, whereas that for odd IGBs are given as n x = m − 1 and n y = p − m + 1.We consider the beam IGB e 4, 2 with  = 2, which is converted to LGB 1,2 and HGB 2,2 when  approaches zero and infinity, respectively.In contrast, IGB o 4, 2 is converted to LGB 1,2 and HGB 1,3 with the same change of .
We first experimentally generate an IGB e 4, 2 as a proof of concept.The calculated phase profile is shown in Figure 2b (top left).An optical microscopy image of the fabricated metasurface with an area of 200 × 200 μm 2 is provided in Figure 2b (top right).The inset shows the corresponding SEM image of metasurfaces.The designed phase profiles are realized with geometric metasurfaces consisting of gold nanorods with spatially variant orientations that can generate PB phase profiles, which are associated with polarization change.The output light includes two parts: the converted part with an additional phase profile change and the nonconverted part without a phase profile change.To filter out the nonconverted part, a linear polarizer (analyzer) and the combination of a quarter waveplate and a linear polarizer are used for the incident light with linear and circular polarization states, respectively.As a proof-of-concept, the feature size of gold nanorods for the metasurfaces is from our previous work, [31]  which can be optimized to have a higher conversion efficiency at a specific wavelength.Alternatively, a dielectric metasurface can be used to dramatically increase the conversion efficiency and suppress the nonconverted part. [28]The nanofabrication details are given in the Experimental Section.Detailed information about the optical setup is provided in Section S1 (Supporting Information).The incident light at 632.8 nm is circularly polarized.The intensity profile of the generated beam is recorded using a CCD camera.The simulated and experimental intensity profiles are given in Figure 2b (bottom left) and Figure 2b (bottom right), respectively.The good agreement between the measured intensity profiles and the theoretical prediction unambiguously shows the generated IGB.It is worth mentioning that the parameter m and (p − m)/2 corresponds to the total number of hyperbolic nodal lines and the number of elliptic nodal lines, respectively.Two hyperbolic nodal lines and one elliptic nodal line are clearly observed in the generated IGB e 4, 2 .More details are available in Section S2 (Supporting Information).
To realize the superposition of IGBs with even and odd modes, the required phase distribution based on the phase multiplexing is written as where a 1 and a 2 are the constants, and  is the initial phase difference.In order to keep the same parity, [42] it is important to note that a 1 ≠ a 2 and p and m must satisfy the relation p = m (see Section S3, Supporting Information).Here, the unique properties of PB phase and phase multiplexing are used to realize the superimposed IGBs. [35]The generation of the PB phase is associated with the change in the circular polarization state, while phase multiplexing is used to calculate the combined phase profile of even and odd IGBs.The simulation and experimental results of the IGB superposition are given in Figure 3.The intensity profile of an individual IGB is provided in Section S4 (Supporting Information).The simulated and experimental intensity profiles are given in the first and second columns, respectively.In Figure 3a, the parameters are given as p = m = 2, a 1 = 0, and a 2 = 2.While in Figure 3b, p = m = 4, a 1 = 2, and a 2 = 0.  = 1/8 and  = 0 are chosen.Dark holes are observed in the resultant beam from the interference between even and odd modes of IGBs.Each dark hole corresponds to a phase singularity.The corresponding phase singularities (3rd column) are observed at the position of dark holes in the simulated phase profiles (highlighted with black circles), which verifies that each dark hole corresponds to a phase singularity.Moreover, there is always a large dark hole with a polygonal structure at the center of the generated beam.IGBs undergo changes in their shape and ellipticity as they propagate through space, whose evolution is influenced by the curvature of the beam and the values of the Ince parameters. [3]The observation plane is located at z = 1 mm in our experiments.The phase distribution in the large dark hole is a combination of multiple mini-phase profiles.There are 2 and 4 mini phase profiles in the big holes.Based on phase change (0 − 2) along the azimuthal direction (clockwise or anticlockwise), the sign of topological charge (TC) can be determined.Therefore, the TCs of central dark holes are −2 and −4, respectively.To verify this point, we experimentally show the interference (4th column) between the superimposed IGBs and a reference spherical beam.Fork-like interference patterns at the positions of dark holes are clearly observed (highlighted with white circles), confirming the existence of phase singularities.The corresponding central dark hole shows the spiral-shaped interference.The total number of spirals is dependent on the absolute value of TC.Based on the TCs, 2 and 4 spirals can be seen in Figure 3a,b, respectively.In addition, the total number of dark holes is 2m + 1.For example, the total number of dark holes in Figure 3a,b are 5 and 9, respectively.Varying the parameter m can alter the number of singularities.The result shows that various phase singularity structures can be generated based on the superposition of IGBs with even and odd modes.
We further manipulate the superimposed IGBs by adding a phase difference between the two IGBs in the design, exhibiting a rotation effect.Figure 4 shows the simulation (1st row) and experimental (2nd row) results with  ranging from 0 to  with an interval of /4.The evolution process of IGB SUP with four different stages is clearly observed.First, a gap slowly appears at the location of each phase singularity when  is increased from 0 to /4.Then, the singularities vanish when  equals /2 and the intensity profile is transformed into four petals.After that, the gap gradually disappears when  is changed from /2 to 3/4.Finally, the intensity profile is restored when  is changed to .It is interesting to note that the sign of TC is reversed.As shown in the corresponding phase profiles (Figure 4 (3rd row)), the value of TC is changed from − 2 at  = 0 to +2 at  = .More details about TC conversion using OAM decomposition are provided in Section S6 (Supporting Information).
The proposed method is further extended to generate the resultant beam through the superposition of IGBs with opposite circular polarization states LP light beam can be decomposed into two beams with opposite CP states and equal components.Figure 5 shows the results of the generated beam under the illumination of LP light.The blue and yellow arrows indicate the transmission axis of the linear polarizer and that of the analyzer, respectively.Figure 5a shows the evolution process of the polarization profile.By gradually rotating the transmission axis of the linear polarizer, the polarization profile of the vector beam is dynamically modulated (Section S7, Supporting Information).The beam contains multiple mini-polarization profiles (as indicated by four red-colored rings).The two mini radial polarization profiles (denoted by R) along the horizontal direction and two mini azimuthal polarization profiles (denoted by A) along the vertical direction are observed in the same beam.Moreover, there is a 2nd order mini polarization profile at the center. [43]Each mini-polarization profile corresponds to a polarization singularity.By continuously varying the transmission axes of the polarizer and analyzer and keeping them perpendicular to each other, we can see the rotation of dark gaps in Figure 5b, which are indicated by white and pink dashed lines for vertical and horizontal polarization profiles, respectively.Keeping directions of the incident linear polarization and the transmission axis of the analyzer perpendicular to each other not only can filter out the nonconverted part, but also can indirectly confirm the polarization profile through the modulated intensity profile with unique features (e.g., gaps).By changing the transmission axis of the polarizer from 0°to 90°with respect to the horizontal direction, two mini radial polarization profiles along the horizontal direction are changed to mini azimuthal polarization profiles, while the two mini azimuthal polarization profiles along the vertical direction are changed to mini radial polarization profiles.To show the robustness of our design, we also demonstrate the resultant beam with eight mini-polarization distributions in Section S8 (Supporting Information).

Discussion
IGBs provide a complete set of eigenmodes in an elliptic coordinate system and the shapes of their intensity profiles are determined by the parameter .By varying the ellipticity, IGBs exhibit continuous transition between LGBs and HGBs in cylindrical and cartesian coordinates, respectively.Phase-only metasurface design is used to demonstrate the concept.To achieve complex amplitude modulation using a metasurface, there are technical challenges in nanofabrication due to the requirement for nanostructures with varying feature sizes.From a practical standpoint, preparing phase-only metasurfaces is more convenient, potentially bringing IGBs one step closer to applications.The good agreement between simulation and experimental results demonstrates the robustness of our metasurface design against fabrication errors.A single metasurface is used to demonstrate the superposition of IGBs with the same circular polarization state, leading to the generation of multiple-phase singularities.The total number of singularities can be controlled by controlling the parameters of even and odd modes, providing a new method to control phase singularities.The initial phase in the metasurface design plays an important role in the modulation of the intensity profiles.We further engineer the phase singularities with the initial phase in the design.This attribute can be used in OAM spectrum manipulation.As an example, Figure S5 (Supporting Information) shows that the OAM spectra of the generated beam can be tuned by altering the initial phase.For instance, when the initial phase is set to 0, the average power of generated IGB is located at TC with a value of −2.In contrast, the average power is shifted to TC with a value of +2 when the initial phase is changed to .Moreover, we can further create helical IGBs (HIGBs) as Unlike a ring of light with a single vortex, a HIGB possesses elliptical intensity rings and contains multiple vortices, as shown in Figure S9 (Supporting Information).The phase is rotated elliptically along the major axis of an ellipse (Section S9, Supporting Information).The direction of rotation can be controlled by the sign in Equation ( 9).This kind of beam carries OAM in elliptical coordinates, which holds promise for applications in particle manipulation and optical tweezers.Furthermore, we demonstrate the resultant beams with multiple polarization singularities.As a proof of concept, we use a liquid crystal polarization rotator to dynamically control the generated polarization profiles (see Section S10, Supporting Information).The marriage between liquid crystals and metasurfaces provides a more sophisticated platform for the dynamic control of vector beams through the superposition of IGBs, opening a new boulevard for applications in many fields such as encryption and optical communications.

Conclusion
In conclusion, we report a metasurface approach to realizing various superpositions of IGBs.The superposition of IGBs with the same circular polarization states can generate beams with multiple phase singularities.Controlling the design parameters of even and odd modes can engineer the number of phase singularities.The introduction of the initial phase in the design can modulate the phase singularity distribution.The generated vector beams through the superposition of IGBs with opposite circular polarization states possess multiple polarization singularities, whose polarization profiles in the local areas can be dynamically modulated by controlling the incident light's linear polarization state.Our study addresses critical challenges in IGB research, including lightweight, compactness, phase singularity manipulation, and dynamic modulation of polarization singularities.These attributes make this technology very promising for diverse applications, spanning singular optics, quantum science, and fundamental physics.The simplicity and robustness of our design hold fundamental significance and can be applied in many fields such as particle trapping, OAM spectrum engineering, and optical communications.

Experimental Section
The transmission-type metasurfaces consist of gold nanorods sitting on a glass substrate (ITO-coated).The electron beam lithography (Raith PIONEER) and lift-off process were used to fabricate the designed metasurfaces.First, 125 nm thick polymethyl methacrylate (PMMA) was spincoated on a glass substrate at 4500 rpm for 60 s.Next, a hotplate was used to bake PMMA film at 95 °C for 3 min.The electron beam lithography was used to define the desired nanorod patterns.An electron beam was used with an accelerating voltage of 30 kV and a beam current of 11 pA.After that, the exposed sample was developed in the mixture of MIBK: IPA (1:3) for 75 s, followed by rinsing in IPA for 60 s.Afterward, an electron beam evaporator was used to deposit a 40 nm thick gold film on a sample.Finally, the metasurface was ready for characterization after the lift-off process in acetone for 4 h.

Figure 1 .
Figure 1.Schematic of a metasurface for realizing the superposition of IGBs.Upon the illumination of RCP light, the superposition of LCP IGBs with even and odd modes can generate multiple phase singularities in the resultant beam.Introducing the initial phase in the design can modulate the phase singularities.The superposition of IGBs with different CPs under the illumination of LP light can generate a vector beam with multiple polarization singularities, which are dynamically modulated by controlling the incident linear polarization.The vector property is characterized through the modulated intensity profile with a linear polarizer (analyzer).The inset shows the change in the intensity profiles with different combinations of the linear polarizer and the analyzer.The transmission axis of the analyzer and that of the first linear polarizer are perpendicular to each other.

Figure 2 .
Figure 2. Transition mechanism of IGBs.a) The transition from IGBs to both LGBs and HGBs.Top: The even mode IGB e 4, 2 is transformed to LGB 1,2 and HGB 2,2 when the elliptic parameter, , approaches zero and infinity, respectively.Bottom: The odd mode IGB o 4, 2 is transformed to LGB 1,2 and HGB 1,3 when  approaches zero and infinity, respectively.b) The calculated phase distribution (top left) for creating IGB e 4, 2 .An optical microscopy image and a scanning electron microscopy (SEM) image of the fabricated metasurface are shown in the top right and in the inset, respectively.Bottom: Simulated (left) and measured intensity profile (right) under the illumination of incident RCP light.

2 Rz
respectively.The parameters p and m should satisfy the relation ( − 1) p − m = 1 and 1 ≤ m ≤ p.  G (z) = arctan ( z z R ) is the Gouy phase and R (z) = z + z is the radius of curvature.The transverse intensity profile of IGB is mainly defined by p, m, and .Since IGBs are the third complete family of orthogonal solutions to the PWE, LGBs and HGBs can be considered as two special cases of IGBs. Figure 2a shows the transition from IGB e,o p, m to both LGB e,o n, l and HGB e,o n x , n y .The indices of LGB e,o n, l (for both even and odd IGBs) are given as l = m, and n = (p−m) 2

Figure 3 .
Figure 3. Superposition of IGBs with even and odd modes.(a) and (b) show the simulated (1st column) and experimental (2nd column) intensity profiles of the different superimposed IGBs.The total number of singularities in (a) and (b) are 5 and 9 due to the different combinations of even and odd modes of IGBs.The 3rd column shows the corresponding phase profile, where the black circles indicate the locations of phase singularities.The 4th column shows the experimental results of the interference pattern, confirming the existence of phase singularities (highlighted with white circles).

Figure 4 .
Figure 4. Effect of the initial phase on the superimposed IGBs.Simulation (1st row) and experimental results (2nd row) of intensity profiles for the metasurface design with the initial phase ranging from 0 to  with an interval of  4 .The 3rd row shows the corresponding phase profiles when the initial phase is changed in the design.

Figure 5 .
Figure 5. Superposition of IGBs with opposite circular polarization states.a) Expected polarization distribution under the illumination of light with different linear polarization directions.Red circles highlight the location of polarization distributions, including mini azimuthal (A) and radial (R) polarization profiles.b) Effect of initial linear polarization state on the modulated intensity profiles.The white dashed lines highlight the directions of dark gaps.Blue and yellow double arrows depict the direction of incident linear polarization and the transmission axis of an analyzer, respectively.They are perpendicular to each other during the experiment.